Question

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives another fusion category, Morita equivalent to the original $\mathcal{C}$.

Can I bound the rank of $\mathcal{C}_\mathcal{M}^*$?

Recall that the rank is the number of isomorphism classes of irreducible objects. Certainly $\mathcal{C}$ and $\mathcal{C}_\mathcal{M}^*$ have the same global dimension, so easily $\operatorname{rank}(\mathcal{C}_\mathcal{M}^*) \leq \operatorname{dim}(\mathcal{C})$. Are there better upper bounds available?

Update: I'm happy to consider all the 'decategorified' data of $\mathcal{C}$ and $\mathcal{M}$, that is, the Grothendieck groups of both, along with the ring and module structures thereon, when trying to come up with an estimate, not just the rank of $\mathcal{C}$.

As examples:

• the Haagerup subfactor gives a Morita equivalence between two fusion categories with ranks 4 and 6, and global dimension $\approx 35.725$
• $\operatorname{Rep}(G)$ and $\text{Vec}_G$ are Morita equivalent, with global dimension $|G|$. Here $\operatorname{rank}(\text{Vec}_G) = |G|$, while when $G$ is non-commutative $\operatorname{rank}(\operatorname{Rep}(G))$ may be much smaller.

Answer 1

If $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent by a pair $({_\mathcal{C}}\mathcal{M}_{\mathcal{D}}, {_\mathcal{D}}\mathcal{N}_{\mathcal{C}})$ of bimodules, then there is a natural map of fusion bimodules ${_D}N \otimes_{C} M_{D} \to {_D}D_D$ that preserves Frobenius-Perron dimension (see section 5.1 of Noah's latest preprint with Pinhas Grossman). So the Frobenius-Perron dimensions of elements of $N \otimes_C M$ (which do not depend on knowing $D$) will give Frobenius-Perron dimensions of elements of $D$. Then I think you may be able to use the known possible small Frobenius-Perron dimensions of objects (from your paper with Noah and Frank Calegari) to determine some nontrivial lower bounds on Frobenius-Perron dimensions of simple objects in $\mathcal{D}$, and thus improve on the global dimension bound.

(Or do arbitrarily small numbers of the form $2 \cos (\pi / n)$ already generate the full ring of real cyclotomic integers? Even if so, this method could at least reduce the bound by 1 for weakly integral categories, although that's not a great improvement.)

Answer 2

In the special case of Vec(G) and Rep(G) there's a lot of results in the literature. Usually the phrasing is in terms of conjugacy classes (e.g. the strongest proved bound as far as I know is Keller's "Finite groups have even more conjugacy classes"). I learned about this from Pavel Etingof when Eric Rowell asked him about a similar question for ranks of centers.

Do I understand correctly though that you're happy to have conditions that involve more than just the rank of C? E.g. if I say wanted to have the full list of dimensions of objects in C as input to the bound would that be a problem?